Four-dimensional regular algebras with point scheme,a nonsingular quadric in P-3

In [22], a class of four-dimensional, quadratic, Artin-Schelter regular alg ebras was introduced, whose point scheme is the graph of an automorphism of a nonsingular quadric in P-3. These algebras are the first examples of qua dratic Artin-Schelter regular algebras whose defining rela- tions are not d etermined by the point scheme and, hence, not determined by the algebraic d ata obtained from the point modules. In this paper, we study these algebras via their line modules. In particular, the set of lines in P-3 that corres pond to left line modules is not the set of lines in P-3 that correspond to right line modules. Our analysis focuses on a distinguished member R-lambd a of this class of algebras, where R-lambda is a twist by a twisting system of the other algebras. We prove that R-lambda is a finite module over its center and that its central Proj is a smooth quadric in P-4.